And I could ask you, “How do you know that? Could you help me prove that?” And we know that if you have two angles that are the same in two different triangles, they are mathematically similar. “What two angles in each of those triangles are the same? Can you help me with that?”

And you might point out to me that all of these angles are the same. In fact they’re all ninety degrees. Because, you tell me, we’re all standing perpendicular to the ground. Even the lamppost is.

You could point out that these angles are the same.

“Why?”

“Because the sun’s rays all strike us at the same angle because we’re all in more or less in the same spot.”

So we have these similar triangles. And I could ask you, “What parts of these triangles are easier to measure than others?”

And we could have that conversation. You might point out that our heights are all known, except the lamppost’s. And our shadows are all fairly easily measured. And at this point, you could solve for that unknown height.

We’ve just gone from the concrete —

— to the very, very abstract.

— you and me — in this process called mathematical abstraction where we formalize the informal. It’s a process that is invaluable to our students, something that mathematicians do all the time. It’s engaging to students and it’s accessible to students at every level.

If I could think of one way to restrict access to math to the already-haves and close it off from the have-nots, I could do no better than to rush as quickly as possible to the highest level of abstraction as possible on this scene. And that’s, of course, how we see this problem in even highly regarded textbooks.

Like right here. I can’t ask you, “What triangles do you see?” Those triangles have already been abstracted.

The problem, as I perceive it, is print. The process that we went through, stepping that out gradually, required ten extra slides in a slidedeck. That cost me a few bits and bytes on a hard drive. It’s nothing. But ten extra printed pages in a textbook. That’s very expensive.

So I’m here today very optimistic about digital curricula and its ability to open that process of abstraction up to all of our students.

That’s not to say this process couldn’t go horribly, horribly wrong.

So I just want to point out here, to close it up and turn it over to you guys, that print is a medium. Same as digital photos. Same as a teacher’s voice. Same as a YouTube video. Same as a podcast. These are all different media. And as we know, the medium is the message. The medium defines and constrains and sometimes distorts the message. The math that can be conveyed in a YouTube video is not the same math that can be conveyed in a digital photo or a podcast or a print textbook.

We’re so enthusiastic here in the Silicon Valley and in this group about technology that disrupts and scales but I think it’s really important to point out here the fundamental misapprehension of this whole process of technology that we have is that there is one monolithic “mathematics” and we are all just innovating around “mathematics.” But those innovations distort what mathematics is. That’s the ball that I urge us all to keep our eye on today. I’m really excited to be here and tease apart those issues with you and take some questions. Thank you.

How will the Common Core Standards paired with the computer assisted adaptive assessments that are envisioned by the Smarter Balanced Consortia change or disrupt middle grades mathematics?

Yeah, I like the Smarter Balanced assessment items, particularly their printed items. And that’s kind of controversial to say, I guess, at a tech conference. But the stuff you can assess online is just different. Someone I admire says, “the computer is not the natural medium for mathematics.” Not yet. There’s no natural language processing. You can’t easily grade automatically and adapt to a written argument the student makes about some figures he says are similar, for instance. So that’s stuff that I would feel sad to lose in our hurry to get to computer adaptive testing.

What kind of support are teachers and schools going to need to transition to Common Core, to make the changes in middle school math? Should the state be providing some support? Should county offices or foundations be providing that support?

Most of the PD that I underwent as a teacher stayed very close to content. It was one or two degrees from how you teach content and, now that we have these practices in the Common Core, that opens up an entirely new PD challenge. The most effective PD I’ve facilitated — and I’ve facilitated some very ineffective PD, I’ll admit that — the effective stuff has always included a large component where we do the practices. Because I think if you’ve taught for thirty years under a particular style of teaching, it has to distort what your perception is of math and how it should be taught. It’s unavoidable, to be steeped in that for so long. So to realign yourself, I imagine, is a very difficult thing. So PD that involves problem solving, involves reasoning, argumentation, that’ll be essential going forward.

Good stuff. One of the real challenges will be how to design activities that are built to go in one direction but are still flexible enough to go in other directions. You say it added 10 slides to the slide deck, but what happens when a kid interjects with some other useful idea?

For your image, a kid might draw along the ground from the base of the pole to a point on the base of Otero (the dorm), then draw a parallel from the top of the pole to the building, and use that to make a good estimate of the height. Or even “Photoshop it” onto a different spot.

Historically in the classroom, the teacher has the flexibility to jump and chase these ideas. Increasingly I see good teachers hamstrung by technology (especially slides): they built a really solid presentation, and darn it they’re going to give their presentation no matter what.

I am hopeful that tablets and other new tech may make it easier to jump out and in or pursue alternate paths through good problems, but am worried that we may instead be further restricting ourselves to presenting and working in a very linear manner. The new media give a lot more options, but how do we make all those options available to students?

Mark Watkins

Taking students out to the lamp changes the math inherent to solving the problem. Maybe that’s a desired result but it would lose any natural need to superimpose triangles on the problem in order to discuss that kind of Geometry; especially if the lamp is short enough to measure in person.

This point is, in effect, what Dan has been trying to say about the medium being the message. Different medium, different math in this case. I have a hard time seeing the natural setting as “better” in terms of teaching the math involved. As for the cross-curricular pollination there’s some merit there but I could provide the same kind of classroom overlap with a photography class by looking at the picture and discussing something other than the math there, as well.

which is based on the experiment that Eratosthenes did over 2200 years ago in measuring the circumference of the earth using only sticks, shadows and brains. Students recreate this by measuring shadows at different places on the globe, share their results and determine the circumference of the earth from that data. Carl Sagan tells the Eratosthenes story engagingly in a segment from the PBS documentary Cosmos available on the website above. The project will start up again in March.

Good stuff. I like Bowen’s idea of going outside if possible. I suggest taking this one more step back. Why would a student believe there is any useful relationship between the heights and lengths? Definitions? Equations? There’s something to notice here about similar triangles, but there’s also something to understand about multiplicative relationships.

I know I’m a bit of a broken record in my previous comments (and this one), but I’m curious what commenters would write as an understanding goal for work on this problem. Here’s my shot at it: Students should understand that there is a constant multiplicative relationship between the legs of similar right triangles. Please suggest improvements!

Finally, Dan’s slides are fantastic and would definitely help a teacher break down the finer points of solving the problem of the height of the lamppost. However, I’m not convinced it would change instruction that much if the goal is to learn how set up and solve proportions using cross-multiplying (as an example of a lesson goal). In order to impact instruction and learning, my learning goal would need to be such that I could clearly see how Dan’s approach would be superior to a less concrete approach. Some of the Common Core Standards are very clear about what students should understand (that a fraction is a number, in grade 3), but others can be interpreted just the same as any other standards document (no mention of ratio being a multiplicative relationship).

Just glad you used an example from the Discovering Geometry text, wish our school had adopted this instead. We (UCSB) do try to incorporate problem solving into our PDs, hoping to understand for ourselves and for our participants what the 8 mathematical practices may look like in the classroom.

John

If the medium changes the message, how do our students shift to meet that medium? The technology doesn’t only change the message (that’s one part of the relationship), it changes and shapes the people relating to it. This is true of text books, iPads, even school as a technology.

In order to understand the technology (a computer let’s say) then we shift ourselves so that our rapport aligns in a way we can understand it.

In this, there is opportunity to enhance and bring us new information, but there is also an opportunity for constriction.

So the distortion isn’t just for mathematics, it can be a distortion of purposeful inquiry and of human exploration. I’m not saying there aren’t opportunities for the use of these tools, it’s just that what they do doesn’t exist in a vacuum. And I bet that PD where you’re actually trying things brings you away from that distortion because any learning happens inside of us during our relationship with the inquiry.

Mansoor

Time: Is the time put into an activity like this worth the benefit? The fact is, most Universities would be thrilled if a graduating senior could just solve a right triangle.

Statistically, I’m far more successful than my peers simply because I manage a classroom well and give my students a consistent, clear routine and test them frequently to make sure they’re learning and retaining skills.

My kids get out of Algebra 1 with skills that allow them to pursue higher math. I don’t evaluate problem solving skills….I don’t think they can be taught. I can simply teach Algebra skills…..they do well in Geometry and Algebra 2.

Your perspective is interesting….not sure if it’s practical. At the high school level, I feel that I’m teaching skills…nothing else.

Am I evil? : )

mr bombastic

I love the subtlety of the initial slide & the gradual development outlined. I don’t really see the need for the other slides other than to illustrate a possible progression for the lesson. Seems like it would be easy enough to project on a whiteboard & draw on that.

I wonder if any students would think to measure the images in the picture. It is interesting that the image of the pole is actually the same height as the image of Dan (1.5 inches or so) even though the pole image appears to be larger. Also interesting how the picture distorts the angle in the triangles by 5 degrees or so.

I agree with Bowen’s point about flexibility & technology. Low tech things like handouts can limit your flexibility as well. Nothing beats a whiteboard for flexibility.

@John – I noted in your link the following in reference to the measurement of earth experiment:

“Our result is x = 39970 km.

If we count the result from the measurements Sint Niklaas – Mikkeli we get x = 41 400 km. The difference is caused mostly by the fact that these cities are not on the same meridian.”

The author of that should be aware that it doesn’t matter if they are not on the same meridian. As long as both measurements are taken when the sun is highest in the sky on the same day (or close to it). They will be measuring at different times since the sun can’t be overhead at the same time on two different meridians.

Like Bowen Kerins pointed out, slideshows can be limiting. The above commenter agrees with that, as well, and adds that really any created piece can be constraining (even low tech handouts).

Our non-profit has built a library of student-created videos that bring math problems to life. These videos are free for teacher use in classrooms and can be a helpful tool in getting students engaged in the subject. It is important to note that these pieces (slides, handouts, or even videos) are aids in the classroom, meant to enhance the lesson and engage students in new and interesting ways. With that in mind, teachers need not be limited in their lessons when students chime in with an idea to move in another (educational) direction.

#10 Mansoor: No offense, but as a college teacher I see lots of students with good “math skills” and no real understanding of the procedures they’ve memorized. It becomes all too evident, when I ask them to do something slightly original with their skills, that having focused for years on passing tests and completing formulaic assignments has prepared them for little else. To paraphrase Paul Lockhart, many a college freshman has come to grief when they discover, after years of being told they were “good at math,” that in fact they are just very good at following directions.

Zack Miller

Just used those exact slides in that exact progression, asking almost those exact questions to two sections of 35 kids. I loved it. I love that entry is easy (buy-in was indeed great; students were engaged; only one student asked “why do I care how tall the post is?”) and, most importantly, I love the level of mathematical abstraction that takes place.

But who did the mathematical abstraction in my class today? I’m glad it wasn’t me and I’m glad it wasn’t a textbook, but, of 70 students, all I know is that it was at least 5 or 6 of them for sure. Maybe 10. Hopefully 30 or 40. Many were there to watch this abstraction unfold like it was – dare I say – a youtube video (granted with a few more “ooooohhh”s than a video usually induces). The math dept wrestled with a lot of ways to present this problem (part-whole class, part-groupwork, etc.); not sure how much that mattered to the 15% of the class that didn’t really get Angle-Angle similarity going into today. They weren’t really in a position to abstract this problem in a meaningful way, any way it was presented.

My point: the content, the progression, the design of this problem and many of Dan’s others are terrific without question (SV needs help from great minds like Dan for this). I have yet to figure out how to get ALL 70 KIDS to really go through this mathematical progression and get what we want out of this problem. Dan, perhaps this is where SV and their computers can help YOU (and all of us!).

Zack Miller writes: ” I have yet to figure out how to get ALL 70 KIDS to really go through this mathematical progression and get what we want out of this problem.”

The key to your answer is examining your phrase “what WE want out this problem.” Unless you include students in the We part you will never really get much more than 5 or 6 really engaged. Unfortunately, that’s business as usual in my opinion. If we want more than marginal improvement in the students performance we have to take seriously what students want to do and include them in the curriculum development. Keith Devlin’s approach with video games is one way to handle curriculum using what kids really want to do as platform. Roger Schank in his latest book “Teaching Minds” has a larger picture solution ideas. As long as our focus is to improve teaching the traditional curriculum things will only improve marginally.. Maybe get 9 or 10 kids more involved out of the 70. Certainly that’s good, but its not good enough as far as Im concerned.

and all of this would be better if…..the kids went outside instead of looking at a picture. That’s even more concrete and will make links to science and the natural world even more explicit.

I don’t think the matter is as simple as, “If kids can interact with the real thing, then they should interact with the real thing.” The decision to go outside and look at the lamppost itself requires an evaluation of pros and cons that is far from straightforward. One of the biggest liabilities of going outside is that you have to ask students to imagine the triangles. You have to wave your hands at them and trace them through the air, hoping their images will stick so you can ask about their characteristics. With the picture, the teacher (or ideally the students) can permanently draw the triangles. There are a dozen more factors just like that one that complicate what jerrid (and others) portray as a simple proposition.

mr bombastic:

I wonder if any students would think to measure the images in the picture. It is interesting that the image of the pole is actually the same height as the image of Dan (1.5 inches or so) even though the pole image appears to be larger.

FWIW, this was intentional. The printed version of this problem has both the lamppost and the person in the same visual plane, allowing students to bypass triangles altogether by measuring the height of the person on the page and the height of the person in real life and scaling that to the lamppost. I tried to fix that.

Zack Miller, who actually used this in class:

But who did the mathematical abstraction in my class today? I’m glad it wasn’t me and I’m glad it wasn’t a textbook, but, of 70 students, all I know is that it was at least 5 or 6 of them for sure. Maybe 10. Hopefully 30 or 40. Many were there to watch this abstraction unfold like it was – dare I say – a youtube video (granted with a few more “ooooohhh”s than a video usually induces).

I can see how this progression resembles a YouTube video. There is a script, but there are two parts to play. The teacher asks useful questions that help the students develop the abstraction and the students answer those questions thoughtfully. Ideally, the teacher ensures the students answer those questions thoughtfully by asking them each to write down their responses, share them with their neighbors, and then share them with the class. The teacher demonstrates that she values thoughtfulness and, over time, the students offer lots of valuable thoughts.

A YouTube video would diminish the script to one part — the teacher’s. That video could begin with the same photo and the teacher could ask all of the same (initial) questions but unless the video offered some way to accept and value student thoughts, the message sent to students will be that their thoughts aren’t valuable.

Again, the message sent to students will be that their thoughts aren’t valuable. The medium has defined the message.

We can remedy this, but only to a degree. The teacher in the video lecture — Sal Khan, let’s say — will never be able to follow-up on student responses, to say, “Whoa. Whoa. I never saw it that way,” when a student suggests something imaginative. But if Sal Khan posed a question like, “How tall do you think that lamppost is?” and gave students a moment to type in a response and then allowed them to see and comment on other students’ responses (a distribution, maybe) we’d be aligning our messages a little: “Your thoughts are somewhat valuable.”

Climeguy:

If we want more than marginal improvement in the students performance we have to take seriously what students want to do and include them in the curriculum development.

I just want to point out how easy it is for folks like Ihor and Keith Devlin (and, hell, me) to disregard standards, assessment, and accountability in our theories of school change. We don’t live within those strictures so they are abstractions to us — just nuisances, really, if they matter at all — and we can imply that the solution is simple: kids should develop the curricula. It takes active, ongoing work for me to tell myself, “the solution is not simple.”

I didn’t realize you would think this is an either/or proposition. If the kids have the iPads (as you imply would be good with your subsequent images) why can’t they take the pictures, and use some sort of app to draw the triangles into the picture rather than you doing it for them. This, to me, would increase the likelihood of the students improving both their problem solving skills, the flexibility of the lesson, and likelihood that students would improve their ability to abstract.

josh g.

If we want more than marginal improvement in the students performance we have to take seriously what students want to do and include them in the curriculum development.

I just want to point out how easy it is for folks like Ihor and Keith Devlin (and, hell, me) to disregard standards, assessment, and accountability in our theories of school change. We don’t live within those strictures so they are abstractions to us — just nuisances, really, if they matter at all — and we can imply that the solution is simple: kids should develop the curricula. It takes active, ongoing work for me to tell myself, “the solution is not simple.”

*Teachers* don’t even have the freedom to define the material covered in the curricula, never mind students.

and comments and I’m convinced that his goal is really what mathematics education should be all about and it doesn’t always have to include symbols. Right now we are at the Standards posing stage. We (adults-who should know better) are wordsmithing to death what the Standards are. Today, it’s called the Common Core. Tomorrow it will be something else. But its a smoke screen because the Standards folks are not saying much about how the Standards are to be achieved. In fact, they leave it up to the individual school districts to figure it out. Keith shares that “Adding It Up: Helping Children Learn Mathematics”

says it all about what needs to be done. What’s needed now is to build curriculums that will actually help students achieve those Standards.

Your work inspires us because you are modeling what teachers can be doing to overcome all the obstacles that textbooks, testing and the conserving nature of schools put in their path. This is great. And I hope you have a standing room only crowd at the NCTM meeting next april. More teachers need to hear your message. (There is no mention of blogging in the conference session descriptions.)

For us old timers who have seen a lot of curriculum reform come and go over the years, time is short so we are encouraging a bold leap to something that makes more sense: curriculums that students actually want to do! This is an incredibly hard challenge (as Keith so vividly points out in his blog), but we now have the technology to make that paradigm shift reality in teaching and learning math. There are plenty of schools that model this idea already. (Here’s just one example of this experiment in progress http://q2l.org/.) We just need to spread the word of what’s possible.

One, I don’t mean to imply I speak for Keith or anybody else. I’m citing a blindspot commonly shared by people who aren’t classroom teachers, myself included.

Two, video game-based learning interests me but it’s another medium and I’m curious — per a recent conversation — how the medium of video-game based learning changes, constrains, or distorts mathematics.

Mark Watkins

I find video games to be interesting as a teaching tool but I’m not convinced it promotes a patient learning environment. There are ways to cheat coded into nearly every single player game on the market; which just screams impatience on the part of the consumer. From my own experience, when a game is frustrating I rarely struggle through the frustration (looking for a walk-through, etc…) and that kind of mentality would probably need to be checked at the door if it’s going to be used for math education as more than a superficial way to drill short tasks.

Zack Miller

@Climeguy: I should have been more clear. Engagement was not at all the issue when I did the activity. The hook, set-up – it all worked. Kids were into it and were committed to using similar triangles to find the height. But, as Dan says, there’s a script. The issue was that with a class of 35, some kids are diverging from the script (drawing triangles every which way, etc.), and so the ones that stick to the script right away are doing the legwork. Yes, I can control that to some degree, but some kids didn’t have a good enough foundation on similarity to get how similarity could be used here.

@Dan: I was hesitant to use the term “youtube video” for fear that you’d see it as an endorsement of that medium for this problem. I completely agree with you; if it’s a video, the script is pre-recorded. The student is a passive observer and that’s not good. This activity was much better than a Youtube video because of the script with two parts to play. Still, though, doing this whole class means I can only hear a few voices out of 35. Doing this individually or in groups causes the “teacher part” of the script to be absent or minimally involved. And this doesn’t even address the biggest unresolved issue for me: what about the kids who aren’t ready for this?

In short, I’m agreeing that the medium is crucial. I agree that a whole-class powerpoint (with pairwork check-ins, etc.) is a much better medium than a youtube video. I just think we can do better, and tech (NOT youtube videos) may have ways to help.

josh g.

Video games as a math learning medium is a weird one. I’ve got a bit of experience on all possible sides – making them, playing them, laughing at how horrible old math-teaching games were – and I’m not sure I’d know how to characterize the medium. Most “math teaching” video games have been so shallow as games that they were little more than computation quizzes. The great examples are rare and sometimes accidental.

Mark, I mostly agree with you – video games in general depend on immediate, continuous feedback to keep the player engaged, which is exactly what a tough abstract math problem does not offer you.

On the other hand, there are puzzle games that break this norm, where players are engaged in extended problem-solving. Or even applications like Foldit where a video game medium has been used to solve actual biochemistry challenges.

However even Foldit has a lot more feedback to the player than the average paper-based textbook trig problem. So, I dunno.

Just a comment isolating the activity from the debate..it’s absolutely awesome. Once again, I feel like my mind has been opened to new mathematical possibilities and I’ll be chasing my shadow everywhere I go.

mr bombastic

I really like the image and the original approach outlined. I am not trying to poke holes in the original lesson, just pointing out a possible extension/stumbling block that might come up.

What intrigues me is that we have these images that appear to be to scale (but are not). I would be pleased if at least a student or two wanted to measure the images as this is a very practical & logical thing to do. If they do measure the images, this problem becomes quite, dare I say it, perplexing.

The ratio of the images of the shadow lengths and the heights are 2 to 1, but the ratio of the actual shadows & heights is 1.6 to 1. Are they going to believe the actual measurements you give them? How are you going to get them to understand the discrepancy between the image ratios and the actual ratios? I think you would need images from other angles to have a chance of convincing them the actual ratio is not 2 to 1. I don’t see the payoff being high enough to go outside for this, but one advantage is they would believe the actual measurements.

Also, the ratio of the images of Dan, the pole, and the woman is 1 : 1 : 0.4, compared to the actual ratio of 1 : 2.1 : .9. The camera shrinks the pole & the woman more than it shrinks Dan. You could estimate some of the scale factors in the picture. Say the woman is 5.5 ft & Dan is 6 ft, and the images are 0.5 inches & 1.25 inches. The scale in the picture is 5.5/0.5 = 11 ft/inch for the woman & 6/1.25 = 4.8 for Dan. The scale for the pole in the picture should be somewhere between 11 ft/inch & 4.8 ft/inch very roughly.

As usual you’ve inspired me. I’ve thought for a long time we need somewhere to crowd-source some inspirational bits of media to work with like this. You do a stunning job of churning out quality materials for everyone. If only we could get more people finding and sharing images and videos.

So in comes Pinterest. I knew there had to be a good use for it and this seems like a match made in heaven.

martisavignali

It is one thing to draw similar triangles and do problems on paper or on screen and use work problems and applications but I think the idea of going outside to be part of the math problem is more effective. Thank you for this useful set of examples.